Kink-equivalence of matrices, spanning surfaces, 4-manifolds, and quadratic forms

TL;DR

This paper introduces the concept of kink-equivalence for matrices, showing that all nonsingular symmetric integer matrices are kink-equivalent to positive or negative definite matrices, with implications for knot theory and 4-manifold topology.

Contribution

It provides a constructive proof that any nonsingular symmetric integer matrix is kink-equivalent to positive or negative definite matrices, establishing bounds on the number of moves needed.

Findings

Every nonsingular symmetric integer matrix is kink-equivalent to a positive-definite matrix.

Every nonsingular symmetric integer matrix is kink-equivalent to a negative-definite matrix.

Implications for knot theory and 4-manifold topology, such as 'alternating up to fake unkinking moves' and classification of 4-manifolds.

Abstract

All checkerboard surfaces for a given knot in $S^3$ are related by isotopy and "kinking" and "unkinking" moves, which change the surfaces' Goeritz matrices like this: $G\leftrightarrow G\oplus [\pm1]=\left[\begin{smallmatrix} G&\mathbf{0}\\ \mathbf{0}^T&\pm1 \end{smallmatrix}\right]$. We call two symmetric integer matrices "kink-equivalent" if they are related by "kinking'' and "unkinking'' moves $G\leftrightarrow G\oplus [\pm1]$ and unimodular congruence. We prove constructively that every nonsingular symmetric integer matrix is kink-equivalent to a positive-definite matrix and to a negative-definite matrix, and we give bounds on the number of moves required. This has several implications, e.g. every knot in $S^3$ is "alternating up to fake unkinking moves" and every simply connected, closed, topological 4-manifold with nonsingular intersection pairing has a positive blow-up that is homeomorphic to a negative blow-up of a positive-definite, simply connected, closed, topological 4-manifold.